profvis demo

n = 1e6
d = tibble(
  x1 = rt(n, df = 3),
  x2 = rt(n, df = 3),
  x3 = rt(n, df = 3),
  x4 = rt(n, df = 3),
  x5 = rt(n, df = 3),
) |>
  mutate(y = -2*x1 - 1*x2 + 0*x3 + 1*x4 + 2*x5 + rnorm(n))

profvis::profvis(lm(y~., data=d))

Benchmarking - bench

d = tibble(
  x = runif(10000),
  y = runif(10000)
)

(b = bench::mark(
  d[d$x > 0.5, ],
  d[which(d$x > 0.5), ],
  subset(d, x > 0.5),
  filter(d, x > 0.5)
))

# A tibble: 4 × 6
  expression                 min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>            <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 d[d$x > 0.5, ]          49.4µs   56.7µs    17251.  238.11KB     46.9
2 d[which(d$x > 0.5), ]   90.7µs    103µs     9595.  268.38KB     48.3
3 subset(d, x > 0.5)      87.5µs  106.8µs     9234.  296.33KB     49.5
4 filter(d, x > 0.5)       287µs    314µs     3061.    1.48MB     18.4

Larger n

d = tibble(
  x = runif(1e6),
  y = runif(1e6)
)

(b = bench::mark(
  d[d$x > 0.5, ],
  d[which(d$x > 0.5), ],
  subset(d, x > 0.5),
  filter(d, x > 0.5)
))

# A tibble: 4 × 6
  expression                 min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>            <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 d[d$x > 0.5, ]          3.62ms   4.08ms     247.     13.4MB     142.
2 d[which(d$x > 0.5), ]   8.12ms   8.93ms     111.     24.8MB     140.
3 subset(d, x > 0.5)      9.46ms  10.12ms      99.0    24.8MB     120.
4 filter(d, x > 0.5)      4.81ms   5.36ms     184.     24.8MB     272.

bench - relative results

summary(b, relative=TRUE)

# A tibble: 4 × 6
  expression              min median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>            <dbl>  <dbl>     <dbl>     <dbl>    <dbl>
1 d[d$x > 0.5, ]         1      1         2.49      1        1.19
2 d[which(d$x > 0.5), ]  2.24   2.19      1.12      1.86     1.17
3 subset(d, x > 0.5)     2.62   2.48      1         1.86     1   
4 filter(d, x > 0.5)     1.33   1.32      1.86      1.86     2.27

parallel

Part of the base packages in R

• tools for the forking of R processes (some functions do not work on Windows)

• Core functions:

  • detectCores

  • pvec

  • mclapply

  • mcparallel & mccollect

detectCores

Surprisingly, detects the number of cores of the current system.

detectCores()

[1] 10

pvec

Parallelization of a vectorized function call

system.time(pvec(1:1e7, sqrt, mc.cores = 1))

   user  system elapsed 
  0.089   0.012   0.100 

system.time(pvec(1:1e7, sqrt, mc.cores = 4))

   user  system elapsed 
  0.126   0.119   0.198 

system.time(pvec(1:1e7, sqrt, mc.cores = 8))

   user  system elapsed 
  0.092   0.165   0.168 

system.time(sqrt(1:1e7))

   user  system elapsed 
  0.018   0.018   0.038 

pvec - bench::system_time

bench::system_time(pvec(1:1e7, sqrt, mc.cores = 1))

process    real 
 61.4ms  60.7ms 

bench::system_time(pvec(1:1e7, sqrt, mc.cores = 4))

process    real 
  157ms   188ms 

bench::system_time(pvec(1:1e7, sqrt, mc.cores = 8))

process    real 
  175ms   207ms 

Cores by size

cores = c(1,4,8,16)
order = 6:8
f = function(x,y) {
  system.time(
    pvec(1:(10^y), sqrt, mc.cores = x)
  )[3]
}

res = map(
  cores, 
  function(x) {
     map_dbl(order, f, x = x)
  }
) |> 
  do.call(rbind, args = _)

rownames(res) = paste0(cores," cores")
colnames(res) = paste0("10^",order)

res

          10^6  10^7  10^8
1 cores  0.005 0.025 0.327
4 cores  0.041 0.161 1.687
8 cores  0.047 0.174 1.361
16 cores 0.055 0.185 1.359

mclapply

Parallelized version of lapply

system.time(rnorm(1e7))

   user  system elapsed 
  0.267   0.006   0.275 

system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 2)))

   user  system elapsed 
  0.320   0.106   0.273 

system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 4)))

   user  system elapsed 
  0.326   0.100   0.179 

system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 8)))

   user  system elapsed 
  0.340   0.135   0.180 

system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 10)))

   user  system elapsed 
  0.354   0.159   0.204 

mcparallel

Asynchronously evaluation of an R expression in a separate process

m = mcparallel(rnorm(1e6))
n = mcparallel(rbeta(1e6,1,1))
o = mcparallel(rgamma(1e6,1,1))

str(m)

List of 2
 $ pid: int 80672
 $ fd : int [1:2] 4 7
 - attr(*, "class")= chr [1:3] "parallelJob" "childProcess" "process"

str(n)

List of 2
 $ pid: int 80673
 $ fd : int [1:2] 5 9
 - attr(*, "class")= chr [1:3] "parallelJob" "childProcess" "process"

mccollect

Checks mcparallel objects for completion

str(mccollect(list(m,n,o)))

List of 3
 $ 80672: num [1:1000000] -1.049 -1.028 0.705 -0.17 1.734 ...
 $ 80673: num [1:1000000] 0.373 0.989 0.259 0.713 0.649 ...
 $ 80674: num [1:1000000] 0.5921 1.2226 0.0756 2.4405 0.0465 ...

mccollect - waiting

p = mcparallel(mean(rnorm(1e5)))
mccollect(p, wait = FALSE, 10) # will retrieve the result (since it's fast)

$`80675`
[1] 0.001344587

mccollect(p, wait = FALSE)     # will signal the job as terminating

NULL

mccollect(p, wait = FALSE)     # there is no longer such a job

NULL

doMC & foreach

Packages by Revolution Analytics that provides the foreach function which is a parallelizable for loop (and then some).

• Core functions:

  • registerDoMC

  • foreach, %dopar%, %do%

registerDoMC

Primarily used to set the number of cores used by foreach, by default uses options("cores") or half the number of cores found by detectCores from the parallel package.

options("cores")

$cores
NULL

detectCores()

[1] 10

getDoParWorkers()

[1] 1

registerDoMC(4)
getDoParWorkers()

[1] 4

foreach

A slightly more powerful version of base for loops (think for with an lapply flavor). Combined with %do% or %dopar% for single or multicore execution.

for(i in 1:10) {
  sqrt(i)
}

foreach(i = 1:5) %do% {
  sqrt(i)   
}

[[1]]
[1] 1

[[2]]
[1] 1.414214

[[3]]
[1] 1.732051

[[4]]
[1] 2

[[5]]
[1] 2.236068

foreach - iterators

foreach can iterate across more than one value, but it doesn’t do length coercion

foreach(i = 1:5, j = 1:5) %do% {
  sqrt(i^2+j^2)   
}

[[1]]
[1] 1.414214

[[2]]
[1] 2.828427

[[3]]
[1] 4.242641

[[4]]
[1] 5.656854

[[5]]
[1] 7.071068

foreach(i = 1:5, j = 1:2) %do% {
  sqrt(i^2+j^2)   
}

[[1]]
[1] 1.414214

[[2]]
[1] 2.828427

foreach - combining results

foreach(i = 1:5, .combine='c') %do% {
  sqrt(i)
}

[1] 1.000000 1.414214 1.732051 2.000000 2.236068

foreach(i = 1:5, .combine='cbind') %do% {
  sqrt(i)
}

     result.1 result.2 result.3 result.4 result.5
[1,]        1 1.414214 1.732051        2 2.236068

foreach(i = 1:5, .combine='+') %do% {
  sqrt(i)
}

[1] 8.382332

foreach - parallelization

Swapping out %do% for %dopar% will use the parallel backend.

registerDoMC(4)

system.time(foreach(i = 1:10) %dopar% mean(rnorm(1e6)))

   user  system elapsed 
  0.296   0.027   0.110 

registerDoMC(8)
system.time(foreach(i = 1:10) %dopar% mean(rnorm(1e6)))

   user  system elapsed 
  0.317   0.045   0.082 

registerDoMC(12)
system.time(foreach(i = 1:10) %dopar% mean(rnorm(1e6)))

   user  system elapsed 
  0.329   0.047   0.074 

furrr / future

system.time( purrr::map(c(1,1,1), Sys.sleep) )

   user  system elapsed 
  0.001   0.000   3.012 

system.time( furrr::future_map(c(1,1,1), Sys.sleep) )

   user  system elapsed 
  0.050   0.007   3.083 

future::plan(future::multisession) # See also future::multicore
system.time( furrr::future_map(c(1,1,1), Sys.sleep) )

   user  system elapsed 
  0.163   0.004   1.454 

Example - Bootstraping

Bootstrapping is a resampling scheme where the original data is repeatedly reconstructed by taking a samples of size n (with replacement) from the original data, and using that to repeat an analysis procedure of interest. Below is an example of fitting a local regression (loess) to some synthetic data, we will construct a bootstrap prediction interval for this model.

set.seed(3212016)
d = data.frame(x = 1:120) |>
    mutate(y = sin(2*pi*x/120) + runif(length(x),-1,1))

l = loess(y ~ x, data=d)
p = predict(l, se=TRUE)

d = d |> mutate(
  pred_y = p$fit,
  pred_y_se = p$se.fit
)

What to use when?

Optimal use of multiple cores is hard, there isn’t one best solution

• Don’t underestimate the overhead cost

• Experimentation is key

• Measure it or it didn’t happen

• Be aware of the trade off between developer time and run time

Statistics and Linear Algebra

An awful lot of statistics is at its core linear algebra.

For example:

• Linear regession models, find

\[ \hat{\beta} = (X^T X)^{-1} X^Ty \]

• Principle component analysis

  • Find \(T = XW\) where \(W\) is a matrix whose columns are the eigenvectors of \(X^TX\).

  • Often solved via SVD - Let \(X = U\Sigma W^T\) then \(T = U\Sigma\).

Numerical Linear Algebra

Not unique to Statistics, these are the type of problems that come up across all areas of numerical computing.

• Numerical linear algebra \(\ne\) mathematical linear algebra

• Efficiency and stability of numerical algorithms matter

  • Designing and implementing these algorithms is hard

• Don’t reinvent the wheel - common core linear algebra tools (well defined API)

BLAS and LAPACK

Low level algorithms for common linear algebra operations

BLAS

• Basic Linear Algebra Subprograms

• Copying, scaling, multiplying vectors and matrices

• Origins go back to 1979, written in Fortran

LAPACK

• Linear Algebra Package

• Higher level functionality building on BLAS.

• Linear solvers, eigenvalues, and matrix decompositions

• Origins go back to 1992, mostly Fortran (expanded on LINPACK, EISPACK)

Modern variants?

Most default BLAS and LAPACK implementations (like R’s defaults) are somewhat dated

• Written in Fortran and designed for a single cpu core

• Certain (potentially non-optimal) hard coded defaults (e.g. block size).

Multithreaded alternatives:

• ATLAS - Automatically Tuned Linear Algebra Software

• OpenBLAS - fork of GotoBLAS from TACC at UTexas

• Intel MKL - Math Kernel Library, part of Intel’s commercial compiler tools

• cuBLAS / Magma - GPU libraries from Nvidia and UTK respectively

• Accelerate / vecLib - Apple’s framework for GPU and multicore computing

OpenBLAS Matrix Multiply Performance

x=matrix(runif(5000^2),ncol=5000)

sizes = c(100,500,1000,2000,3000,4000,5000)
cores = c(1,2,4,8,16)

sapply(
  cores, 
  function(n_cores) 
  {
    flexiblas::flexiblas_set_num_threads(n_cores)
    sapply(
      sizes, 
      function(s) 
      {
        y = x[1:s,1:s]
        system.time(y %*% y)[3]
      }
    )
  }
)